We were discussing the Torsion
or twisting moment, Shear stress produced in a circular shaft subjected to torsion and Torque transmitted by a circular solid shaft in our previous
posts.
Now we are going further to start a new topic i.e. Torque
transmitted by a circular hollow shaft with the help of this post.
So, what is torsion or twisting moment?
A shaft will said to be in torsion, if it will be
subjected with two equal and opposite torques applied at its two ends.
When a shaft will be subjected to torsion or
twisting moment, there will be developed shear stress and shear strain in the
shaft material.
We will discuss here one case of a hollow circular
shaft which will be subjected to torsion and we will secure here the expression
for maximum torque transmitted by a hollow circular shaft.
We have following information from above figure
Ro = Outer radius of the hollow circular shaft
Ri = Inner radius of the hollow circular shaft
Do = Outer diameter of the hollow circular shaft
Di = Inner diameter of the hollow circular shaft
dr = Thickness of small elementary circular ring
r = Radius of the small elementary of circular ring
q = Shear stress at a radius r from the centre of
the hollow circular shaft
τ = Maximum shear stress at outer surface of shaft
dA = Area of the small elementary of circular ring
dA = 2П x r x dr
Shear stress, at a radius r from the centre, could
be determined as mentioned here
q/r = τ / Ro
q = τ x r/ Ro
Turning force due to shear stress at a radius r from
the centre could be determined as mentioned here
dF = q x dA
dF = τ x r/ Ro x 2П x r x dr
dF = τ/ Ro x 2П r2dr
Twisting moment at the circular elementary ring
could be determined as mentioned here
dT = Turning force x r
dT = τ/ Ro x 2П r3dr
Total torque could be easily determined by
integrating the above equation between limits Ri and Ro.
Therefore total torque transmitted by a hollow
circular shaft will be given by following formula
Do you have suggestions? Please write in
comment box.
We will now discuss the Principal
planes and principal stresses, in the category of strength of material, in our
next post.
Reference:
Strength of material, By R. K. Bansal
Image Courtesy: Google
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